If i have a shaft & i'm applying a driving torque D at one end & at other end there is resisting torque R due to bearing friction, etc. Then if D>R, i have net torque = moment of inertia times angular acceleration. Since the ang. accln is constant with respect to time, will the ang. speed of shaft keep increasing till infinity? If the answer is yes, then why doesnt this happen to motors & engines @ low or no load? Another doubt is that by conservation of energy we have input power= output power + losses, so if i am giving finite input power, the output power has to be finite. Since rotational kinetic energy of shaft is half * MI * square of angular velocity, angular velocity cannot be infinite as that would make output power infinite, right? Please help this is so confusing

In motors, as the angular velocity approaches the synchronized velocity, which depends on the frequency of the electric power and also on the number of poles in the winding, the torque decrease. in absolute no-lode, the torque becomes zero. This happens because when the rotor is rotating with the same speed as the magnetic field is rotating, no current is induced in the rotor.

For engines also there is a system to reduce the fuel to keep the speed low, unless you press the gas pedal.

Also the motor torque tends to decrease as speed increases. Note, that may not be true for low speeds with some types of motor (e.g. a car gasoline or diesel engine), but it will be true when the speed gets high enough, because any real motor can only deliver a limited amount of power, and power = torque x RPM.

The ideal-world answer is that the rpm wouldn't increase to infinity (as we're limited by the speed of light), but would indeed continue increasing to a high level IF we can continue applying the torque.

Realistically we either can't keep up with it for the various mentioned reasons, or, perhaps more likely, we are simply limited by the mechanical strength of the spinning object. Stresses due to spinning increase as a function of RPM2, so they get very high very fast. The limit to how fast you can safely spin is the redline, which is usually kept from being exceeded by a mechanical or electronic governor.

If the redline DOES get exceeded, the spinning object, such as a motor, can fail catastrophically.

In a motor, if the rpm exceeds the rotation speed of the magnetic field, the shaft experiences a negative torque which acts like a break on the motor. Of course the motor would be acting like a generator then.